python 实现 viterbi 算法

viterbi 算法是一种动态规划算法，用于解决马尔可夫模型的解码问题。马尔可夫模型是一种概率模型，用于随机过程中的状态转移和观测。viterbi 算法可以用于解决各种问题，例如语音识别、自然语言处理和生物信息学。

viterbi 算法的基本思想是，通过计算每个状态在每个时间步长的概率，来找到最有可能的状态序列。具体来说，viterbi 算法的步骤如下：

1. 初始化：初始化每个状态在初始时间步长的概率。
2. 迭代：对于每个时间步长，计算每个状态的概率，并更新每个状态的概率。
3. 终止：在最后一个时间步长，选择概率最大的状态序列。

viterbi 算法的时间复杂度为 O(n^2)，其中 n 是状态的数量。

下面是一个简单的 python 实现的 viterbi 算法：

def viterbi(observations, states, transition_probabilities, emission_probabilities):
  """
  Computes the most likely state sequence given a sequence of observations.

  Args:
    observations: A list of observations.
    states: A list of states.
    transition_probabilities: A matrix of transition probabilities.
    emission_probabilities: A matrix of emission probabilities.

  Returns:
    A list of the most likely state sequence.
  """

  # Initialize the trellis.
  trellis = [[0 for _ in range(len(states))] for _ in range(len(observations))]
  backpointers = [[None for _ in range(len(states))] for _ in range(len(observations))]

  # Initialize the first column of the trellis.
  for i, state in enumerate(states):
    trellis[0][i] = transition_probabilities[0][i] * emission_probabilities[i][observations[0]]

  # Fill in the rest of the trellis.
  for t in range(1, len(observations)):
    for i, state in enumerate(states):
      trellis[t][i] = max([trellis[t - 1][j] * transition_probabilities[j][i] for j in range(len(states))]) * emission_probabilities[i][observations[t]]
      backpointers[t][i] = np.argmax([trellis[t - 1][j] * transition_probabilities[j][i] for j in range(len(states))])

  # Backtrack to find the most likely state sequence.
  state_sequence = []
  i = np.argmax([trellis[-1][i] for i in range(len(states))])
  state_sequence.append(i)
  for t in range(len(observations) - 1, 0, -1):
    i = backpointers[t][i]
    state_sequence.append(i)

  state_sequence.reverse()

  return state_sequence

本文针对经典的天气-海藻湿度案例，使用 python 实现了简单的 viterbi 解码算法，大家可以对照理论来理解代码。